122 ANALYTICAI MECHANICS It is interesting to note that tie motions in the two directions are independent. The gravi ational acceleration does not affect the constant velocity atong the £-axis, while the motion along the y-axis is the san<.e as if the body were projected vertically with a velocity VQ sin a. The projectile virtually rises a distance of v0t sin a on account of its initial vertical velocity, and falls a distance f gt2 on account of the gravitational acceleration. THE PATH. — The equation of the path may be obtained by eliminating t between equations (7) and (8). This gives which is the equation of a parabola. THE TIME OF FLIGHT. — When the projectile strikes the ground its y-coordinate is zero. Therefore substituting zero for y in equation (8) we get for the time of flight a THE RANGE. —The range, or the total horizontal distance "covered by the projectile, is found by replacing t in equation (7) by the value of 5P, or by letting y = 0 in equation (9). By •either method we obtain n 2 V02 sin a COS a Jtl =--------------'----------- !sin2r ^ 'Since VQ and g are constants the value of R depends upon a. It is evident from equation (11) that R is maximum when .sin 2 a = 1, or when a = |. The maximum range is, therefore, (12)